Multistep Hermite collocation methods for solving Volterra integral equations
A new method is presented to obtain numerical solutions of a class of nonlinear Volterra integral equations of the form \[ y(t)=g(t)+\int_0^t K(t,s,y(s))ds,\quad t\in I:=[0,T],\tag{\(*\)} \] where \(g:I\to \mathbb R\) is sufficiently smooth and \(K:D\times \mathbb R\to \mathbb R\), with \(D:=\{(t,s):0\leq s\leq t\leq T\}\), is continuous and is Lipschitzian in \(y\). In this paper the authors use (\(*\)) and \[ y'(t)=g'(t)+\int_0^t K_t(t,s,y(s))ds+K(t,t,y(t)),\quad t\in I, \] so that the approximate solution in each subinterval depends on the values of the approximated solution and its first derivative in the fixed number \(r\) of previous time steps and also the values of the approximate solution and its first derivative in the \(m\) collocation points. This technique enables them to find higher order methods with extensive stability region. This new method, which is called multistep Hermite collocation method, has the uniform convergence order \(2m+2r\) for \(r\) steps and \(m\) collocation points. Some numerical examples are presented to show the efficiency of the method.
scientific article; zbMATH DE number 6031711
scientific article